Hammerstein equation

A non-linear integral equation of the type

$$\phi(x)+\int\limits_a^bK(x,s)f[s,\phi(s)]ds=0,\quad a\leq x\leq b,$$

where $K(x,s)$ and $f(x,s)$ are given functions, while $\phi(x)$ is the unknown function. Named after A. Hammerstein [1], who considered the case where $K(x,s)$ is a symmetric and positive Fredholm kernel, i.e. all its eigen values are positive. If, in addition, the function $f(x,s)$ is continuous and satisfies the condition

$$|f(x,s)|\leq C_1|s|+C_2,$$

where $C_1$ and $C_2$ are positive constants and $C_1$ is smaller than the first eigen value of the kernel $K(x,s)$, the Hammerstein equation has at least one continuous solution. If, on the other hand, $f(x,s)$ happens to be a non-decreasing function of $s$ for any fixed $x$ from the interval $(a,b)$, Hammerstein's equation cannot have more than one solution. This property holds also if $f(x,s)$ satisfies the condition

$$|f(x,s_1)-f(x,s_2)|\leq C|s_1-s_2|,$$

where the positive constant $C$ is smaller than the first eigen value of the kernel $K(x,s)$. A solution of the Hammerstein equation may be constructed by the method of successive approximation (cf. Sequential approximation, method of).

References

[1]	A. Hammerstein, "Nichtlineare Integralgleichungen nebst Anwendungen" Acta Math. , 54 (1930) pp. 117–176
[2]	F.G. Tricomi, "Integral equations" , Dover, reprint (1985)
[3]	M.M. Vainberg, "Variational methods for the study of nonlinear operators" , Holden-Day (1964) (Translated from Russian)
[4]	M.A. Krasnosel'skii, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964) (Translated from Russian)
[5]	N.S. Smirnov, "Introduction to the theory of integral equations" , Moscow-Leningrad (1936) (In Russian)